If the value of $\int {x\sin x = - x cosx + \alpha } $ , then the value of $\alpha $ is :
A. $\sin x$ +C
B. $\cos x$ +C
C. C
D. None of these

Question

If the value of $\int {x\sin x =  - x cosx + \alpha } $ , then the value of $\alpha $ is :A. $\sin x$ +CB. $\cos x$ +CC. CD. None of these

Vedantu Content Team · Accepted Answer

Hint: Whenever there is a product of the two functions, then integration by parts or partial integration should be used. For instance, if  and   are the function of x, the integration of $$\int {u.vdx}  = u\int {vdx}  - \int {\left[ {\left( {\dfrac{{du}}{{dx}}} \right)\int {vdx} } \right]} dx$$ , where $u = $ first function and $v = $ second function.Complete step by step solution:$\int {x\sin x =  - x cosx + \alpha } ......(1)$From the integration it is clear that it is a product of two functions of x. Therefore, integration by parts should be used to evaluate the integral.Let us assume x as the first function and  as the second function.$$I = x\int {\sin xdx}  - \int {\left[ {\left( {\dfrac{{d\left( x \right)}}{{dx}}} \right)\int {\sin xdx} } \right]dx} ......(2)$$ The integration of $\int {\sin xdx =  - co} sx$  and the differentiation of $x = 1$ , put the value in equation (2) $I =  - x\cos x + \int {\cos xdx} ......(3)$ The integration of $$\int {\cos xdx = \sin } x$$  put the value in equation (3) $I =  - x\cos x + \sin x + C......(4)$ Where, C is a constant and known as constant of integration. Comparing equation (4) and (1), it is clear that The value of $\alpha  = \sin x + C$ Hence, the correct option is (A).Note: The choice of the first and second function should be in accordance with the thumb rule, which says that choose the function in the order of ILATE where I is the inverse trigonometric function, L is the logarithmic function, A is the algebraic function, T is the Trigonometric function and E is the exponential function. For instance in $\int {x\cos x} $ x is the first function and  is the second function because x is the algebraic function and cosine function is the trigonometric function. So the preference for first function is given to x.

If the value of $\int {x\sin x = - x cosx + \alpha } $ , then the value of $\alpha $ is :
A. $\sin x$ +C
B. $\cos x$ +C
C. C
D. None of these

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If the value of $\int {x\sin x = - x cosx + \alpha } $ , then the value of $\alpha $ is :A. $\sin x$ +CB. $\cos x$ +CC. CD. None of these

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If the value of $\int {x\sin x = - x cosx + \alpha } $ , then the value of $\alpha $ is :
A. $\sin x$ +C
B. $\cos x$ +C
C. C
D. None of these